Kamal Khuri-Makdisi's generals
My advanced topics were algebraic number theory and algebraic geometry.
The committee consisted of Shimura (head), Faltings, and Nelson. The questions
were, in order:
Explain Galois theory, and give an example (I gave the splitting field of
x^3 - 2 over Q). What can you say about the case of x^(2^n) - 2?
Given a sequence of continuous functions from [0,1] -> R that tends to 0
pointwise, do their integrals tend to 0? Give a counterexample.
Take a decreasing sequence of continuous functions on [0,1] that tends to 0
pointwise. Is the convergence uniform? (I didn't know, so was told to guess.
The answer is yes; this is a theorem of Dini's.) Prove it.
If f : C -> C is analytic and bounded, what happens? What if f is entire
but only Re f is bounded?
Talk about doubly periodic functions on C. (I mentioned the Weierstrass P
and P'- functions, and stated without proof the differential equation that
they satisfy.) Prove that the sum of the residues of such a function in a
period parallelogram is 0.
Talk about the Fourier transform of a function on R. Show that the transform
of an L^1 function is continuous and tends to 0 at infinity. What about the
transform of an L^2 function? (I said it could be extended to an isometry
of L^2, but wasn't asked to prove this.) State the inversion formula. When
does it hold?
Talk about the analytic continuation of your favorite transcendental function.
(I wrote down 'log' but was interrupted.) No, that's too simple. What about
the zeta function? Talk about zeta and L-functions in arbitrary number fields
(define them; to what sort of function can they be analytically continued?
Almost no proofs were required, except for really easy things).
State the main theorems of class field theory. (They stopped me halfway
through.)
What is a scheme?
Define the genus of a curve (in a couple of ways). Explain how to compute it
using the Hilbert polynomial. Why is this the same as the dimension of the
space of global 1-forms? For a coherent sheaf F on P^r associated to the
graded module M, why is the Hilbert polynomial H(n) (:= Euler characteristic
of F(n)) equal to the dimension of M_n for n >> 0? (I sketched a proof.)
Compute the genus of the curve y^2 = x^5 - x by using Hurwitz' formula.
(The curve is singular, so really consider its normalization. The function
x gives a degree 2 morphism from this curve to P^1, so count the number of
ramification points and apply the formula.)
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The exam lasted 2 hours.